Minimum variance carrier recovery with increased phase noise tolerance

ABSTRACT

A method of data symbol recovery in a coherent receiver of an optical communications system includes processing data symbol estimates detected from a received optical signal, and determining recovered symbol values from the processed data symbol estimates. The recovered symbol values belong to a symbol constellation having a predetermined asymmetry. Processing the data symbol estimates compensates phase noise that is greater than one decision region of the symbol constellation. A coherent receiver of an optical communications system includes a module configured to process data symbol estimates detected from a received optical signal and a decision circuit configured to determine recovered symbol values from the processed data symbol estimates. The recovered symbol values belong to a symbol constellation having a predetermined asymmetry. Processing the data symbol estimates compensates phase noise that is greater than one decision region of the symbol constellation.

TECHNICAL FIELD

The present invention relates generally to optical communicationsystems, and in particular to systems and methods for data symbolrecovery in a coherent receiver.

BACKGROUND

In optical communication systems that employ coherent optical receivers,the modulated optical signal received at the coherent receiver is mixedwith a narrow-line-width local oscillator (LO) signal, and the combinedsignal is made incident on one or more photodetectors. The frequencyspectrum of the electrical current appearing at the photodetectoroutput(s) is substantially proportional to the convolution of thereceived optical signal and the local oscillator (LO), and contains asignal component lying at an intermediate frequency that contains datamodulated onto the received signal. Consequently, this “data component”can be isolated and detected by electronically filtering and processingthe photodetector output current.

The LO signal is typically produced using a semiconductor laser, whichis typically designed to have a frequency that closely matches thefrequency of the laser producing the carrier signal at the transmitter.However, as is known in the art, such semiconductor lasers exhibit afinite line width from non-zero phase noise. As a result, frequencytransients as high as ±400 MHz at rates of up to 50 kHz are common. Thisfrequency offset creates an unbounded linear ramp in the phasedifference between the two lasers. In addition, many such lasers oftenexhibit a line width of the order of 1 MHz with a Lorentzian spectralshape. As a result, even if the transmitter and LO lasers were tooperate at exactly the same average frequency, a phase error linewidthof about ±2 MHz can still exist. This Lorentzian spectrum creates aphase variance that grows linearly with time, and the initial phasedifference is random, so over the lifetime of operation of the opticalconnection the phase error is unbounded.

As is known in the art, data is typically encoded in accordance with aselected encoding scheme (e.g. Binary Phase Shift Keying (BPSK);Quadrature Phase Shift Keying (QPSK), 16-Quadrature Amplitude Modulation(16-QAM) etc.) to produce symbols having predetermined amplitude andphase. These symbols are then modulated onto an optical carrier fortransmission through the optical communications system to a receiver. Atthe receiver, the received optical signal is processed to determine themost likely value of each transmitted symbol, so as to recover theoriginal data.

As is known in the art, a frequency mismatch or offset Δf, andindependent phase noise between the transmitter and LO laser appears asa time-varying phase θ of the detected symbols, relative to the phasespace of the applicable encoding scheme. This variation of the symbolphase θ is exacerbated by phase non-linearities of the opticalcommunications system, and in particular, cross-phase modulation (XPM).The symbol phase θ is unbounded, in that it tends to follow arandom-walk trajectory and can rise to effectively infinite multiples of2π. Because the symbol phase θ is unbounded, it cannot be compensated bya bounded filtering function. However, unbounded filtering functions aresusceptible to cycle slips and symbol errors, as will be described ingreater detail below.

Applicant's U.S. Pat. No. 7,606,498 entitled Carrier Recovery in aCoherent Optical Receiver, which issued Oct. 20, 2009, teachestechniques for detecting symbols in the presence of a frequency mismatchbetween the received carrier (that is, the transmitter) and the LOlaser. The entire content of U.S. Pat. No. 7,606,498 is incorporatedherein by reference. In the system of U.S. Pat. No. 7,606,498, aninbound optical signal is received through an optical link 2, split intoorthogonal polarizations by a Polarization Beam Splitter 4, and thenmixed with a Local Oscillator (LO) signal by a conventional 90° hybrid8. The composite optical signals emerging from the optical hybrid 8 aresupplied to respective photodetectors 10, which generate correspondinganalog signals. The analog photodetector signals are sampled byrespective Analog-to-Digital (A/D) converters 12 to yield multi-bitdigital sample streams corresponding to In-phase (I) and Quadrature (Q)components of each of the received polarizations.

The format and periodicity of the SYNC bursts may conveniently beselected as described in U.S. Pat. No. 7,606,498. In each of theembodiments illustrated in FIGS. 2A and 2B, the optical signal includesnominally regularly spaced SYNC bursts 14 embedded within a stream ofdata symbols 16. Each SYNC burst 14 has a respective predeterminedsymbol sequence on each transmitted polarization. In the embodiment ofFIG. 2A, two orthogonal symbol sequences are used in each SYNC burst 14;each symbol sequence being assigned to a respective transmittedpolarization. FIG. 2B illustrates an alternative arrangement, in whicheach of the I and Q components of each transmitted polarization isassigned a respective orthogonal symbol sequence.

Returning to FIG. 1, from the A/D converter 12 block, the I and Q samplestreams of each received polarization are supplied to a respectivedispersion compensator 18, which operates on the sample stream(s) tocompensate chromatic dispersion. The dispersion-compensated samplestreams appearing at the output of the dispersion compensators 18 arethen supplied to a polarization compensator 20 which operates tocompensate polarization effects, and thereby de-convolve transmittedsymbols from the complex sample streams output from the dispersioncompensators 18. If desired, the polarization compensator 20 may operateas described in Applicant's U.S. Pat. No. 7,555,227 which issued Jun.30, 2009. The entire content of U.S. Pat. No. 7,555,227 is incorporatedherein by reference. The polarization compensator 20 outputscomplex-valued symbol estimates X′(n) and Y′(n) of the symbolstransmitted on each polarization. These symbol estimates include phaseerror due to the frequency offset Δf between the Tx and LO frequencies,laser line width and Cross-phase modulation (XPM). The symbol estimatesX′(n) and Y′(n) are supplied to a carrier recovery block 26 (see FIG.1), which performs carrier recovery and phase error correction, andsymbol determination. Two known carrier recovery and symboldetermination techniques are described below.

In the system of U.S. Pat. No. 7,606,498 each SYNC burst is used todetermine an initial phase error value γ₀, which is used to calculate aninitial phase rotation κ₀ for the start of processing the next block ofdata symbols. Once the SYNC burst has been processed, the receiverswitches to a data directed mode, during which the phase rotation isupdated at predetermined intervals and applied to successive data symbolestimates X′(n) and Y′(n) to produce corresponding rotated data symbolestimates X′(n)e^(−jκ(n)) and Y′(n)e^(−jκ(n)). The decision value X(n),Y(n) of each transmitted symbol can be determined by identifying thedecision region in which the rotated symbol estimate lies, and thesymbol phase error γ calculated and used to update the phase rotation.

Applicant's U.S. Pat. No. 8,315,528 which issued Nov. 20, 2012 teaches azero-mean carrier recovery technique in which two or more SYNC burstsare processed to derive an estimate of a phase slope ψ indicative of thefrequency offset Δf between the transmit laser and the Local Oscillator(LO) of the receiver. The phase slope ψ is then used to compute a phaserotation κ(n) which is applied to each symbol estimate X′(n), Y′(n) toproduce corresponding rotated data symbol estimates X′(n)e^(−jκ(n)),Y′(n)e^(−jκ(n)) which can then be filtered to remove XPM and find thedecision values X(n), Y(n) of each transmitted data symbol. The entirecontent of U.S. Pat. No. 8,315,528 is incorporated herein by reference.

The processes described in U.S. Pat. Nos. 7,606,498 and 8,315,528 areunbounded, and thus can compensate unbounded symbol phase κ. However,both of these techniques assume that each rotated symbol estimateX′(n)e^(−jκ(n)) and Y′(n)e^(−jκ(n)) lies in the correct decision regionof the symbol phase space. This means that when the symbol phase error γbecomes large enough (e.g. ≥π/4 for QPSK, or ≥π/2 for BPSK) the rotatedsymbol estimate will be erroneously interpreted as lying in a decisionregion that is adjacent to the correct decision region. When this occursin respect of an isolated symbol estimate, the resulting “symbol error”will be limited to the affected symbol. On the other hand, where asignificant number of symbol errors occur in succession, the receivermay incorrectly determine that a “cycle-slip” has occurred, and resetits carrier phase to “correct” the problem. Conversely, the receiver mayalso fail to detect a cycle slip that has actually occurred. This canresult in the erroneous interpretation of a large number of symbols.FIGS. 3A and 3B illustrate this problem.

FIG. 3A illustrates a Quadrature Phase Shift Keying (QPSK) symbolconstellation comprising four symbols (A-D) symmetrically disposed in aCartesian symbol space defined by Real (Re) and Imaginary (Im) axes. Inthe symbol space of FIG. 3A, each quadrant corresponds with a decisionregion used for determining the respective decision value for eachrotated symbol estimate. A rotated symbol estimate 28 is a complex valuecomposed of Real and Imaginary components, which can therefore be mappedto the symbol space, as may be seen in FIG. 3A. It is also convenient torepresent the rotated symbol estimate 28 as a polar coordinate vectorhaving a phase θ and magnitude M, as shown in FIG. 3A.

As may be seen in FIG. 3A, the assumption that the rotated data symbol28 estimate lies in the correct decision region means that the symbolphase error γ is calculated as the angle between the rotated data symbolestimate 28 and the nearest symbol of the encoding scheme (symbol B inFIG. 3A). As may be seen in FIG. 3B, the calculated phase error γ iszero when the phase θ corresponds with a symbol of the constellation,and increases linearly as the phase θ approaches a boundary between twodecision regions. However, as the phase θ crosses a decision boundary(at θ=0, ±π/2, and ±π in FIGS. 3A and 3B), there is a discontinuity inthe calculated phase error γ. For example, as the phase θ increasesthrough the decision boundary at π/2, the calculated phase error γreaches +π/4, and then jumps to −π/4, which is π/2 away from the correctphase error. This discontinuity increases the probability of makingsubsequent symbol errors, and can contribute to the occurrence of cycleslips. Once a cycle slip has occurred, subsequently received symbolswill be incorrectly decoded until the problem has been detected andrectified.

An alternative frequency and phase estimation technique known in the artis the Viterbi-Viterbi algorithm, in which the Cartesian coordinatesymbol estimates X′(n) and Y′(n) are raised to the fourth power todetermine the phase rotation value that has the greatest probability ofoccurring and then these values are filtered using Cartesian averaging.The resulting phase rotation is then divided by four and applied to thereceived samples to try to determine the most likely decision valuesX(n), Y(n) of each transmitted data symbol. This approach suffers alimitation in that dividing the phase estimate by four also divides the2π phase ambiguity by four, meaning that if incorrectly resolved thisambiguity causes a π/2 cycle slip. This technique can providesatisfactory performance in cases where the phase errors are dominatedby a small frequency offset between the TX and LO lasers and moderatelaser line widths. However, in the presence of XPM, this approachbecomes highly vulnerable to producing cycle slips.

In some cases, the above-noted problems can be mitigated by use of asufficiently strong Forward Error Correction (FEC) encoding scheme, butonly at a cost of increased overhead, which is undesirable.

Techniques for carrier recovery that overcome limitations of the priorart remain highly desirable.

SUMMARY

Disclosed herein are techniques for carrier recovery and data symboldetection in an optical communications system.

Accordingly, an aspect of the present invention provides method of datasymbol recovery. An optical signal is modulated by a transmitter using amodulation scheme comprising a symbol constellation having apredetermined asymmetry and detected at a receiver. Phase errorestimates corresponding to data symbol estimates detected from thereceived optical signal are calculated. A phase rotation is calculatedbased on the phase error estimates, using a filter function, and thephase rotation applied to at least one data symbol estimate to generatea corresponding rotated symbol estimate. The phase error estimates modelthe asymmetry of the symbol constellation, such that the computed phaserotation can compensate phase noise that is greater than one decisionregion of the symbol constellation.

BRIEF DESCRIPTION OF THE DRAWINGS

Representative embodiments of the invention will now be described by wayof example only with reference to the accompanying drawings, in which:

FIG. 1 is a block diagram schematically illustrating a coherent opticalreceiver, known from U.S. Pat. No. 7,606,498;

FIGS. 2A and 2B schematically illustrate respective alternative signalformats known from U.S. Pat. No. 7,606,498;

FIGS. 3A and 3B schematically illustrate relationships between rotatedsymbol estimates computed in accordance with the prior art, and anencoding constellation known in the prior art;

FIG. 4 is a chart schematically illustrating relationships betweensymbol phase error and maximum likelihood phase error in accordance witha representative embodiment of the present invention;

FIG. 5 is a flow-chart illustrating principal steps in a carrierrecovery process in accordance with a representative embodiment of thepresent invention;

FIG. 6 is a block diagram illustrating principal elements in a carrierrecovery block implementing methods in accordance with a representativeembodiment of the present invention;

FIG. 7 is a block diagram illustrating the carrier recovery block ofFIG. 6 in greater detail;

FIGS. 8A and 8B schematically illustrate operation of an embodiment inwhich an optical signal is modulated in accordance with a modulationscheme comprising an asymmetrical constellation scheme;

FIGS. 9A and 9B illustrate a second asymmetrical constellation usable inembodiments of the present invention; and

FIGS. 10A and 10B illustrate a third asymmetrical constellation usablein embodiments of the present invention.

It will be noted that throughout the appended drawings, like featuresare identified by like reference numerals.

DETAILED DESCRIPTION

The present invention exploits the observation that the probability thata symbol estimate lies in any given decision region of the applicableencoding scheme is a maximum when the symbol estimate lies on, or verynear, the corresponding symbol of the encoding scheme, and decreaseswith increasing distance from the symbol, but is not zero at theboundary with an adjacent decision region.

In very general terms, the present disclosure provides techniques inwhich the symbol estimates are processed to compute a probabilisticphase error φ that reflects both the symbol phase error γ and theprobability that the symbol estimate is lying in the correct decisionregion. The probabilistic phase error φ is then filtered and used tocompute a minimum variance phase rotation κ(n) applied to eachsuccessive symbol estimate.

An advantage of the carrier recovery technique disclosed herein is thatit models the overall statistical performance of the opticalcommunication system within the carrier recovery algorithm itself. Thisis an improvement over prior art techniques which model specificdistortions (such as frequency offset, line width or XPM) and thenrelying on a strong FEC to correct erroneous symbols due to otherdistortions (such as ASE) in a post-processing step. This improvement isbeneficial in that it allows the FEC to correct more errors from othersources, and thereby improves the performance of the opticalcommunications system for subscriber traffic.

In the prior art examples of FIGS. 1-3, the phase error γ is computed asthe angle between a rotated symbol estimate 28 and the nearest symbol inthe applicable encoding scheme, and used to compute the phase rotationapplied to successive symbol estimates. If desired, such feed-backcarrier recovery techniques (and variants thereof) may be used inembodiments of the present invention. Alternatively, feed-forwardtechniques may be used to compute an estimate of the phase error γ.

It is contemplated that embodiments of the present invention may beimplemented in a coherent optical receiver using any suitablecombination of hardware and software. For very high speed applications,hardware implementations, for example using one or more FieldProgrammable Gate Arrays (FPGAs) or Application Specific IntegratedCircuits (ASICs) will normally be preferable, but this is not essential.

The probability that the rotated symbol estimate 28 lies in the correctdecision region is a function of the location of the symbol estimate 28in the symbol space of the applicable encoding scheme. Referring back toFIG. 3A, for any given value of the vector magnitude M, the probabilitythat the symbol estimate 28 lies in the correct decision region is amaximum when the calculated phase error γ equals zero, and decreaseswith increasing absolute value of the phase error γ. At the boundarybetween adjacent decision regions, the symbol estimate has an equal(non-zero) probability of being in either decision region. Accordingly,the probabilistic phase error φ will tend to be proportional to thephase error γ for values of γ close to zero, and will be zero at aboundary between adjacent decision regions, as may be seen in FIG. 4.

Furthermore, for any given symbol phase θ, the probability that thesymbol estimate 28 lies in the correct decision region increases withincreasing values of M. This may be understood by recognizing that agiven magnitude of additive noise (eg Amplified Spontaneous Emission(ASE)) affecting the rotated symbol estimate 28 will have aproportionately greater impact on the symbol phase θ at smaller valuesof M than at positions farther away from the origin. Accordingly, forany given value of the phase θ, the probabilistic phase error φ willtend to be proportional to the magnitude M of the symbol estimate. Thisresults in a family of probabilistic phase error curves for differentvalues of M. FIG. 4 illustrates a family of five representativeprobabilistic phase error curves, for values of M=m1, m2, m3, m4 and m5,respectively. In this way, the knowledge of M is incorporated in to theprobabilistic estimate.

In some embodiments, it is desirable for the calculation of theprobabilistic phase error φ to minimize the variance of each phaseestimate, including the variance due to symbol errors (the L₂ norm). Insuch cases, the probabilistic phase error φ can be calculated as theexpected value of the random variable representing the phase error γ,given the knowledge supplied (primarily symbol phase θ and magnitude M).In other embodiments, it may be desirable to minimize the peak value ofthe absolute error between the probabilistic estimate and the actualvalue of the random variable representing the phase error γ, (the L_(∞)norm), or the integral of the error, (the L₁ norm), or other similarprobability operations. In general, the probabilistic estimate φ ofphase-error γ can be defined, using a variety of different metrics ordifferent operators, on the “conditional” probability density functionof phase-error, conditioned on the received symbol phase θ and magnitudeM. The examples mentioned above (L₁, L₂, L_(∞) norms) are some specificuseful operators derived from the conditional probability densityfunction of phase-error, but this list is not exhaustive.

A phase error estimate that attempts to minimize a norm in this mannergives improved performance compared to the prior art estimation methodsthat try to estimate the mode of the probability density, i.e. the phasewith greatest probability density, with some level of quantization.

Various methods may be used to compute the probabilistic phase error φfor any given rotated symbol estimate 28. For example, two or moreprobabilistic phase error curves may be explicitly defined as a functionof the symbol phase θ (using any suitable technique) for respectivedifferent values of the magnitude M, and then known interpolationtechniques used to compute the probabilistic phase error φ for themagnitude M(n) and phase θ(n) of each rotated symbol estimate 28. In analternative arrangement, a look-up table may be used to define a mappingbetween a set of predetermined values of the symbol phase θ andmagnitude M, and the probabilistic phase error φ. In operation, eachrotated symbol estimate 28 can be processed to determine its phase θ(n)and magnitude M(n), which can then be used as an index vector suppliedto the input of the look-up-table, which outputs the correspondingprobabilistic phase error φ(n). In some embodiments, rounding may beused to reduce the size of the look-up table. For example, consider acase where the phase θ(n) and magnitude M(n) of each symbol estimate arecomputed with 8-bits of resolution. These two values may be concatenatedto produce a 16-bit index vector supplied to the look-up-table, in whichcase the look-up-table will require at least 2¹⁶=65 kilo-bytes ofmemory. If, on the other hand, the phase θ(n) and magnitude M(n) arerounded to 3 bits resolution each (e.g. by taking the 3 most significantbits), then the size of the look-up-table may be reduced to 2⁶=64 bytesof memory. Minimizing the size of this table can be important because itis accessed at the sample rate of the receiver, which can be tens ofbillions of samples per second.

In still further embodiments, the above-described techniques may beused, but with the probabilistic phase error φ defined as a function ofthe calculated symbol phase error γ rather than its phase θ. Thisarrangement is advantageous, in that it permits the probabilistic phaseerror φ(n) to be computed to a higher precision, because the symbolphase error γ only spans the angular width of a single decision region(i.e. π/2 for QPSK) whereas the symbol phase θ spans the entire 2π phasespace.

It will be further understood that the above-noted techniques can bereadily extended to encoding schemes, such as 16-QAM, for example, inwhich the decision regions are delimited by both phase θ and magnitudeM, or to other codes such as multi dimensional codes, differentialcodes, and codes including both polarizations.

To further enhance accuracy of the above method for phase errorestimation, the calculation of the probabilistic phase error φ(n)estimate may incorporate useful metrics and operators of a conditionalprobability density function of phase error, conditioned on respectivesymbol phase and magnitude values of a plurality of successive symbolestimates. However, the complexity of the method increases exponentiallywith the number of symbol estimates considered in the probabilisticphase error estimation method. As an example, the method can use therespective phase and magnitude of symbol estimates on X and Ypolarization and calculate the L₁ norm (or any other suitable operator)of the conditional probability density function of phase-error,conditioned on M_(x) and M_(y) (received magnitudes on eachpolarization) and θ_(x) and θ_(y) (received phase values on eachpolarization). To further reduce complexity of the method, orequivalently the size of the look-up table, it is possible to usevarious functions of the received magnitude and phase, rather than thephase and magnitude values themselves. As an example, in the abovescenario, the probabilistic phase-error estimate (based on L₁ or L₂metric of conditional probability density function) may be conditionedon values of A and B, defined as A=M_(x)M_(y) and B=(θ_(x) moduloπ/2)+(θ_(y) modulo π/2). The parameters A and B can be computed to anysuitable precision, and rounded (or quantized) to 3 bits so that a64-byte memory LUT can be used to generate the probabilistic phase errorestimate φ(n).

FIG. 5 shows a flow-chart illustrating principal operations of a carrierrecovery block implementing minimum variance carrier estimation inaccordance with a representative embodiment of the present invention. Ata first step (step S2), a symbol estimate received from the polarizationcompensator 20 (FIG. 1) is received and processed (at step S4) todetermine its polar coordinate magnitude M(n) and phase θ(n) (or phaseerror γ(n), if desired). The polar coordinate values are used todetermine a respective probabilistic phase error φ(n) value for thesymbol (at step S6), which is then filtered (at step S8) to minimize theeffects of XPM, Additive Gaussian Noise (AGN), and symbol errors. Thus,in this example, a probabilistic phase error estimate is produced fromindividual symbol estimates, and then the time series of probabilisticphase error estimates is filtered to minimize the variance in the timeseries. In some embodiments, a Wiener filter is used. In otherembodiments, an approximation of a Wiener filter may yield satisfactoryresults. For example, as the Wiener filter generally is a low passfilter, it can be approximated as a moving average filter. The filteroutput represents a minimum variance estimate of the symbol phase error,and is used (at step S10) to compute a corresponding minimum variancephase rotation κ(n) which compensates frequency offset Δf, laser linewidth and XPM. This minimum variance phase rotation κ(n) is then appliedto the symbol estimate (at step S12) to yield a corresponding rotatedsymbol estimate in which the residual phase error is random andcomparatively small. As a result, rotated symbol estimates can beprocessed using known methods to determine a decision value representingthe most likely value of each transmitted symbol. An advantage of thisapproach is improved noise tolerance in the receiver.

FIG. 6 schematically illustrates a representative carrier recovery block26 which implements methods in accordance with a representativeembodiment of the present invention.

In the embodiment of FIG. 6, the carrier recovery block 26 is dividedinto two substantially identical processing paths 30; one for eachtransmitted polarization. Each processing path 30 receives a respectiveoutput of the polarization compensator 20, and outputs recovered symbolsof its respective transmitted polarization. Each processing path 30includes a frequency error estimator 32, a phase noise estimator 34, aphase rotator 36, and a decision circuit 38. In general, each phaserotator 36 uses an estimate of the phase noise generated by the phasenoise estimator 34 to compute and apply a phase rotation κ(n) to thesymbol estimates received from the polarization compensator 20. Thephase-rotated symbol estimates X′(n)e^(−jκ(n)) and Y′(n)e^(−jκ(n))generated by the phase rotators 36 are then processed by the decisioncircuits 38 to generate the recovered symbol values X(n) and Y(n).Representative embodiments of each of these blocks will be described ingreater detail below.

FIG. 7 schematically illustrates the carrier recovery block 26 of FIG. 6in greater detail. It will be noted that only the X-Polarizationprocessing path 30 _(X) is illustrated in FIG. 7, it being understoodthat a substantially identical arrangement will be provided for theY-Polarization processing path 30 _(Y). The carrier recovery 26 block ofFIG. 7 is configured for two operating modes, namely: a “training” modewhile processing a SYNC burst 14; and a “data directed” mode whilerecovering transmitted data symbols 16. In the training mode, thecorrelation values output by the polarization compensator 20 aresupplied to the frequency error estimator 32, which computes a localslope ψ as an estimate of the frequency error due to the frequencyoffset Δf between the transmit laser and the LO and independent laserline width. In the embodiment of FIG. 7, the SYNC burst correlationvalues output by the polarization compensator 20 are accumulated (at 40)to average the correlation across at least a portion of the SYNC burst14. The I and Q components of the averaged correlation value are thenused to compute a phase error estimate

${\Delta\phi}_{SYNC}\mspace{14mu}{\tan^{- 1}\left( \frac{Q}{I} \right)}$of the SYNC burst at 42. This SYNC burst phase error estimate representsthe average phase error of the symbols comprising the SYNC burst,relative to the ideal phase of those symbols, as determined by theencoding format (e.g. PSK, QPSK, 16-QAM etc.) of the optical signal.

As may be appreciated, the symbol estimates of each SYNC burst containphase errors due to frequency offset Δf, laser linewidth, and XPM.Computing an average phase error of each SYNC burst has an effect oflow-pass filtering the individual phase errors of the SYNC burst symbolsat 44, and so tends to reduce the effects of laser phase noise and XPM.A further reduction in the effects of laser phase noise and XPM can beobtained by low-pass filtering the respective phase error estimatesΔϕP_(SYNC)(i) of two or more successive SYNC bursts (i=1 . . . m) tocompute the local slope ψ.

Once the SYNC symbols 14 have been processed, the receiver switches tothe data directed mode, during which the phase noise is computed andused to rotate data symbol estimates, and the resulting rotated symbolestimates X′(n)e^(−jκ(n)) and Y′(n)e^(−jκ(n)) processed by the decisioncircuits 38 to generate the recovered symbol values X(n) and Y(n). Thisoperation will be described in greater detail below.

In general, the phase noise estimator 34 uses the symbol estimatesoutput from the polarization compensator 20, and the local slope ψoutput from the frequency estimator 32 to compute a minimum varianceestimate of the phase noise due to frequency offset Δf, independentlaser line width, and XPM.

As may be seen in FIG. 7, during each clock cycle (m), a set of Nsuccessive symbol estimates X′(i) from the polarization compensator 22are supplied to a converter 46 (which may be configured as a CORDICconverter known in the art), which processes the Real(Re) andImaginary(Im) components of each symbol estimate X′(i) to obtain thecorresponding polar coordinate (magnitude (M) and phase (θ)) values. Thephase θ is corrected or the unbounded phase ramp due to the frequencyoffset Δf, as represented by the local slope ψ output by the frequencydetector 32. The resulting corrected phase θ_(CORR) is then supplied toa phase detector 48, which outputs an estimate of the phase error valueγ(n) of the corresponding symbol estimate.

If desired, respective phase error value γ(n) can be accumulated over aset of N successive symbol estimates, and used to compute a Minimum MeanSquare Estimate (MMSE) phase error, which can then be used in subsequentprocessing as described below. It will be noted that the above-describedtechniques for estimating the phase error γ(n) are examples offeed-forward methods, since the phase error γ(n) is estimated withoutreference to the decision value X(n) output by the decision block 38. Inalternative embodiments, the feed-back computation methods described inU.S. Pat. Nos. 7,606,498 or 8,315,528 may be used to compute theestimated symbol phase error γ(n).

Referring again to FIG. 7, the respective magnitude and phase errorvalue γ(n) are then supplied to a look-up-table 50, which is configuredto output a probabilistic phase error φ(n) value based on the receivedsymbol magnitude and phase error value γ(n). The probabilistic phaseerror φ(n) value output from the LUT 50, is then filtered (at 52) toremove residual noise due to AGN and XPM. As mentioned previously, aWiener filter may be used for this purpose. Alternatively,approximations of a low pass filter function, such an computing arunning average over the respective probabilistic phase error φ(n)values obtained for set N successive symbol estimates may be used. Thefilter output represents a minimum variance estimate of the phase noiseΔγ(n), which is then passed to the phase rotation block 36.

In general, the phase rotation block 36 computes and imposes a phaserotation κ(n) which compensates phase errors of the corresponding symbolestimates X′(n) and Y′(n), due to frequency offset, laser line width andXPM. For example, each successive value of the phase rotation κ(n) maybe computed using a function of the form:κ(n+1)=κ(n)+μ₁ψ+μ₂Δγ(n+1)where the scaling factors μ₁ and μ₂ may be programmable, and define thephase adjustment step sizes for each successive data symbol estimatewithin the data block. The first order phase rotation term μ₁ψcompensates the unbounded phase rotation due to the frequency offset Δfbetween the transmit and LO lasers and independent laser line-width. Thesecond order phase rotation term μ₂Δγ(n+1) represents the minimumvariance phase noise estimate computed as described above, and isupdated at the symbol rate. Integrating the result over the frequencyrange of the optical signal yields the variance of the residual XPM andlaser noise. This is only mathematically valid because the unboundedphase error contributions are compensated by the first order phaserotation term μ₁ψ.

Taken together, the first and second order terms μ₁ψ and μ₂Δγ(n+1)provide an estimate of the incremental phase change Δκ between then^(th) and (n+1)^(th) symbols. Accumulating this incremental value Δκfor each successive data symbol yields the updated phase rotation κ(n+1)at 53.

Applying the phase rotation κ(n) to each symbol estimate X′(n) and Y′(n)at 54 yields rotated symbol estimates X′(n)e^(−jκ(n)) andY′(n)e^(−jκ(n)) in which the unbounded phase rotation due to thefrequency offset Δf between the Tx and LO lasers, independent laserline-width, and XPM have been removed. The streams of rotated symbolestimates X′(n)e^(−jκ(n)) and Y′(n)e^(−jκ(n)) will therefore exhibit aminimum variance phase error, with short period phase excursions dueprimarily to AGN.

If desired, the decision block 38 may operate to determine the decisionvalues X(n) and Y(n) representing the most likely transmitted symbols,in a manner as described in U.S. Pat. No. 7,606,498.

In the foregoing examples, each probabilistic phase error φ(n) value iscalculated from a corresponding complex valued symbol estimate.Alternatively, each probabilistic phase error φ(n) may be calculatedfrom multiple symbols (as noted above) or more intricate symbolestimates. Linear filtering of the probabilistic phase error estimatesis advantageous for minimizing complexity of implementation. However,nonlinear filtering operations could be performed, if desired. Cartesianfiltering could be used, but it generally suffers performancedegradations in the presence of XPM. For simplicity of description, thecarrier recovery block 26 is described as having a single processingpath 30 for each polarization. However, to enable implementation at highspeeds it may be desirable to implement parallel processing paths,and/or approximate operations. For example, one minimum variancerotation value κ(n) may be computed and then applied to a set of N (e.g.four, eight, or sixteen) successive symbol estimates.

The probabilistic phase error calculation described above uses only thelinearly processed symbol estimates, in polar coordinates. However,other information and other processing could be used. For example, errorprobabilities from parity bits, forward error correction, errorprobability estimation, or turbo equalization could also be used.Nonlinear polarization or phase compensation could be applied.

Furthermore, the probabilistic phase error calculation described aboveuses all of the symbol estimates output from the polarizationcompensator 20 (FIG. 1). However, if desired, the probabilistic phasecalculation can be a nonlinear function of a plurality of samples. For achannel with highly correlated XPM, an advantageous example of this isto consider the respective phase errors of a set of N (e.g. N=16)consecutive symbol estimates, and discard the M (e.g. M=2) outliers inthe set. The subsequent calculations and filtering are a function of theremaining N-M samples in this set, and other such sets. Alternatively,nonlinear filtering of a sequence of probabilistic phase estimates couldcomprise an emphasis operation such as Median Filtering where a subsetof the estimates is selected. For simplicity of implementation, theseexamples use a choice of inclusion or exclusion of individual samples.However, more subtle methods of emphasis and de-emphasis could be usedif desired. For example: multiplication of an estimate by a multi-bitweight, or trellis selection where sequences of candidate decodings ofthe samples are considered and one sequence is selected.

It is desirable that the minimum variance rotation κ(n) applied to agiven symbol estimate not be derived from the phase and magnitude ofthat symbol estimate, but rather only from the phase and magnitudevalues for other symbol estimates which have respective phase errorsthat are correlated to the phase error of the given symbol. In otherwords, the optimal filter function applied to the time series ofprobabilistic phase error φ(n) values (FIG. 5 at S8) has an impulseresponse that is zero at time zero. For simplicity of implementation,one may choose the sub-optimal approximation of including someinformation derived from the given symbol in its own rotation estimate,if that information has been diluted with information from many othersymbols.

In the foregoing description, the calculation of the probabilistic phaseerror φ(n) is based on the symbol estimate represented by its polarcoordinate values of magnitude M and phaseθ (or phase error γ(n)).However, it is contemplated that a mathematically equivalent calculationmay be performed using the Cartesian coordinate representation of thesymbol estimate. In such a case, the step of computing the magnitude Mand phase θ at FIG. 5, step S4 is omitted. Instead, the Real andImaginary components of the symbol estimate may be adjusted tocompensate for the frequency offset Δf, and then combined to form aninput vector to a look-up-table to obtain the probabilistic phase errorφ(n). The look-up table may output the probabilistic phase error φ(n) ineither polar or Cartesian coordinate form, as desired. In embodiments inwhich the Cartesian coordinate form of the probabilistic phase errorφ(n) is used, it will be desirable to convert the minimum variance phasenoise Δγ(n) output from the filter (FIG. 7 at 52) to polar coordinaterepresentation for computation of the phase rotation κ(n), orequivalently scale to unit radius and conjugate multiply. Cartesianaveraging could be used in the filtering, but that generally suffersperformance degradations in the presence of XPM.

The methods disclosed herein have at least two advantages over knownViterbi-Viterbi phase and frequency estimation algorithms. Inparticular, the present technique considers the statistical propertiesof the additive noise (such as ASE) and the additive phase noise in thephase estimation. In other words, the phase-detector output depends onthe probability density function of phase-error and additive noise (suchas ASE) and any source of noise/distortion in the system. In contrast,known Viterbi-Viterbi algorithms yield a phase-error estimate that isindependent from statistics of the system and channel. For example, ifthe standard deviation of additive noise (e.g. Amplified SpontaneousEmission (ASE)) becomes much smaller (compared with the phase noise),the output of the phase noise estimator 34 (FIG. 6) will have lessdependency on the information of the received symbol's magnitude value(M(n)), independent of the phase-error γ(n) of any given symbolestimate. However, prior art methods such as Viterbi-Viterbi estimatethe phase error based on the phase difference to the decided symbol, sothe estimated phase error will vary on a per-symbol basis, independentof the statistics of the optical communication system. Furthermore, thepresently disclosed methods can take in to account the probability ofmaking mistake in deciding the actual transmitted symbol. In otherwords, the calculation of the conditional probability function ofphase-error also can consider the probability that the transmittedsymbol is not the symbol closest to the symbol estimate. In contrast,prior art methods such as Viterbi-Viterbi assume that the transmittedsymbol is the symbol closest to the symbol estimate which causesincreased phase variance or risk of cycle slips.

FIGS. 8A and 8B illustrate an embodiment in which a modulation schemeused to modulate an optical signal in a transmitter comprises anasymmetrical symbol constellation. As may be seen in FIG. 8A, theconstellation is a QPSK constellation with π/2 asymmetry. In this case,symbols A and C are modulated with an amplitude M=m1 that is smallerthan the amplitude M=m2 of symbols B and C. Following the methodsdescribed above, the probabilistic phase error φ can be computed tomodel the asymmetry of the constellation. As may be seen in FIG. 8B,this results in a set of overlapping phase error probability functionsin which the probabilistic phase error φ for each value of M tends to beproportional to the phase error γ for values of γ close to zero, andwill be zero at a phase error of γ=±π/2.

Accordingly, in the embodiment of FIGS. 8A and 8B, the probabilisticphase error φ is capable of detecting and at least partially correctingphase error γ that is greater than the respective decision region of anygiven symbol of the constellation. It will be appreciated that, ascompared to a conventional symmetrical QPSK symbol constellation, theembodiment of FIGS. 8A and 8B is significantly more tolerant of phasenoise. The asymmetry can be used to make a control loop resilient,“self-righting”, to phase transients that are greater than π/2 and lessthan π, and therefore more resistant to cycle slips.

Other asymmetrical symbol constellations can be designed to achievedesired levels of phase noise tolerance, and thus resistance to cycleslips. For example, FIGS. 9A and 9B illustrate a constrained phasesymbol constellation of the type known from Applicant's U.S. Pat. No.8,983,309 which issued Mar, 17, 2015, the entire content of which ishereby incorporated herein by reference. As may be seen in FIG. 9A, theillustrated constellation has 2π-asymmetry, and a modulation phase θthat is constrained to a phase range θ_(min) to θ_(max) spanning lessthan 4π. As may be seen in FIG. 9B, this results in a phase errorprobability function in which the probabilistic phase error φ varieslinearly with (and may, in fact, equal) the phase error γ for values ofdetected modulation phase θ above θ_(max) and below θ_(min). With thisarrangement, the probabilistic phase error φ is capable of detecting andcorrecting phase noise up to 2π. In addition, at least one of thesymbols lying adjacent the θ_(min) to θ_(max) deadband (in theillustrated example, symbols A and D) can be unconditionally detected,even at very high values of the phase error γ. This means that, even ifa cycle slip does occur, it can be unconditionally detected, andcorrected, upon detecting either one of the symbols lying adjacent theθ_(min) to θ_(max) deadband. It will be appreciated that this providesan automatic mechanism for both correcting cycle slips and limiting thenumber of errored data symbols produced by a cycle slip.

FIGS. 10A and 10B illustrate an asymmetrical BPSK symbol constellationwith π asymmetry. In this case, symbol A is modulated with an amplitudeM=m1 that is larger than the amplitude M=m2 of symbol B. Following themethods described above, the probabilistic phase error φ can be computedto model the asymmetry of the constellation. As may be seen in FIG. 10B,this results in a set of overlapping phase error probability functionsin which the probabilistic phase error φ for each value of M tends to beproportional to the phase error γ for values of γ close to zero, andwill be zero at a phase error of γ=±π. Consequently, the probabilisticphase error φ is capable of detecting and at least partially correctingphase error γ that is greater than the conventional ±π/2 decision regionof either symbol of the constellation. It will be appreciated that, ascompared to a conventional symmetrical BPSK symbol constellation, theembodiment of FIGS. 10A and 10B is significantly more tolerant of phasenoise. The asymmetry can be used to make a control loop resilient,“self-righting”, to phase transients that are greater than π.

FIGS. 8-10 show embodiments of 2-dimensional constellations with varyingdegrees of asymmetry. It is contemplated that higher dimensional (i.e.N-dimensional constellations, with N equal to 2 or more) asymmetricalconstellations can be designed, and the asymmetry modeled in theprobabilistic phase error φ to achieve increased phase noise toleranceand cycle slip resilience using techniques directly analogous to thosedescribed herein.

For clarity of description, the foregoing example embodiments use theprobabilistic phase error φ to calculate a phase rotation that is usedto rotate the symbol estimates to compensate phase noise prior todecoding the result. It is functionally equivalent to rotate the frameof reference of the decoder rather than symbol estimates, or to rotateboth in such a manner that the sum of the rotations yields the desiredeffect. Phase rotations can be implemented via a CORDIC rotator, bycomplex multiplication in Cartesian coordinates, by addition in polarcoordinates, or any other substantially equivalent operations. Ifdesired, phase rotations can be combined with other operations such asscaling.

In the foregoing example embodiments, the detected phase error γ is usedto compute a probabilistic phase error estimate φ that is used todetermine the phase rotation κ(n) that compensates phase noise. It isadvantageous for the phase error estimate φ to be probabilistic, such asa scalar value based upon an expected value, but this is not essential.Other approximations such as piece-wise linear or sinusoidal, and otherfunctions, minimizations, or optimizations can be used to define a phaseerror estimate φ that retains an aspect which models an asymmetry of thesymbol constellation. The phase error estimate φ could bemultidimensional, or complex, such as having dimensions corresponding topolarization or time. These dimensions can be independent, or havecontrolled amounts of correlation.

In the examples shown above there is one phase error estimate φ fromeach data symbol estimate. Alternatively there can multiple phase errorestimates from one or more data symbol estimates, such as via differingfunctions. One phase error estimate could be the average, composite, orvector of a plurality of data symbol estimates. In general there can beN phase estimates corresponding to a set of M data symbol estimates.

In feed-forward methods, a constellation asymmetry can be used to guidethe phase unwrap operation to improve the probability of making correctchoices of unwraps. This guidance is an aspect of the phase errorestimate φ. This method reduces the probability of a cycle slip due to apersistently incorrect unwrap.

Although the invention has been described with reference to certainspecific embodiments, various modifications thereof will be apparent tothose skilled in the art without departing from the spirit and scope ofthe invention as outlined in the claims appended hereto.

What is claimed is:
 1. A method of data symbol recovery in a coherentreceiver of an optical communications system, the method comprising:computing probabilistic phase error estimates based on data symbolestimates detected from a received optical signal; computing a phaserotation based on the probabilistic phase error estimates; applying thecomputed phase rotation to at least one data symbol estimate to generatea corresponding rotated symbol estimate; and determining a recoveredsymbol value from the rotated symbol estimate, wherein the recoveredsymbol value belongs to a symbol constellation having a predeterminedasymmetry with respect to phase rotation, and the computed phaserotation compensates phase noise that is greater than any decisionregion of the symbol constellation.
 2. The method according to claim 1,wherein computing the probabilistic phase error comprises calculating anexpected value of phase noise based on a conditional probability densityfunction, the probability density function being conditioned on phaseand magnitude of at least one data symbol estimate.
 3. The methodaccording to claim 2, wherein the conditional probability densityfunction models the asymmetry of the symbol constellation.
 4. The methodaccording to claim 1, wherein computing the probabilistic phase errorestimates comprises, for each data symbol estimate: calculatingrespective polar coordinate symbol phase and magnitude values of thedata symbol estimate; calculating a phase error of the data symbolestimate, based on the calculated symbol phase; and calculating therespective probabilistic phase error estimate based on the calculatedphase error and the symbol magnitude.
 5. The method according to claim4, wherein calculating the phase error comprises: adjusting the detectedsymbol phase to compensate frequency offset between a transmit laser anda local oscillator of the coherent receiver, to generate a correctedsymbol phase; and calculating the phase error between the correctedsymbol phase and a nearest symbol of the symbol constellation.
 6. Themethod according to claim 1, wherein computing the phase rotationcomprises: filtering the probabilistic phase error estimates calculatedfor a time series of successive data symbol estimates to obtain aminimum variance phase error; and calculating a minimum variance phaserotation using the minimum variance phase error.
 7. The method accordingto claim 6, wherein filtering the probabilistic phase error estimatescomprises any one or more of: applying a Wiener filter to theprobabilistic phase error estimates; and computing a running averageover the probabilistic phase error estimates.
 8. The method according toclaim 6, wherein calculating the minimum variance phase rotationcomprises accumulating, for each successive data symbol estimate, acorresponding incremental phase rotation corresponding to μ₁ψ+μ₂Δγ(n+1),where μ₁ and μ₂ are scaling factors defining a phase adjustment stepsize for each successive data symbol estimate, ψ is a phase slopeindicative of unbounded phase error due to frequency offset, and Δγ(n+1)is the minimum variance phase error.
 9. The method according to claim 6,wherein filtering the probabilistic phase error estimates comprises theoperation of emphasis of a subset of the probabilistic phase errorestimate values being filtered.
 10. The method according to claim 1,wherein the probabilistic phase error estimates are representative of aphase error of each data symbol estimate and a probability that eachdata symbol estimate is in a correct decision region of the symbolconstellation.
 11. A coherent receiver of an optical communicationssystem, the coherent receiver comprising: a phase noise estimatorconfigured to compute probabilistic phase error estimates based on thedata symbol estimates detected from a received optical signal; a phaserotator configured to compute a phase rotation based on theprobabilistic phase error estimates and to apply the computed phaserotation to at least one data symbol estimate to generate acorresponding rotated symbol estimate; and a decision circuit configuredto determine a recovered symbol value from the rotated symbol estimate,the recovered symbol value belonging to a symbol constellation having apredetermined asymmetry with respect to phase rotation, wherein thecomputed phase rotation compensates phase noise that is greater than anydecision region of the symbol constellation.
 12. The coherent receiveraccording to claim 11, wherein the phase noise estimator is configuredto compute the probabilistic phase error by calculating an expectedvalue of phase noise based on a conditional probability densityfunction, the probability density function being conditioned on at leastone data symbol estimate.
 13. The coherent receiver according to claim11, wherein the phase noise estimator comprises: a converter configuredto calculate respective polar coordinate symbol phase and magnitudevalues of each data symbol estimate; a phase detector configured tocalculate a phase error of each data symbol estimate, based on thecalculated symbol phase; and a Minimum Mean Square Error processorconfigured to calculate the probabilistic phase error estimate based onthe calculated phase error and the symbol magnitude.
 14. The coherentreceiver according to claim 13, wherein the Minimum Mean Square Errorprocessor comprises a look-up table.
 15. The coherent receiver accordingto claim 13, further comprising: a frequency correction block configuredto adjust the detected symbol phase to compensate frequency offsetbetween a transmit laser and a local oscillator of the coherentreceiver, to generate a corrected symbol phase; and wherein the phasedetector is configured to calculate the phase error between thecorrected symbol phase and a nearest symbol of the symbol constellation.16. The coherent receiver of claim 11, further comprising a filter blockconfigured to filter minimum variance phase error estimates calculatedfor a time series of successive data symbol estimates to obtain aminimum variance phase error estimate.
 17. The coherent receiver ofclaim 16, wherein the phase rotator is configured to calculate a minimumvariance phase rotation using the minimum variance phase error estimate.18. The coherent receiver of claim 17, wherein the phase rotator isconfigured to compute the minimum variance phase rotation byaccumulating, for each successive data symbol estimate, a correspondingincremental phase rotation corresponding to μ₁ψ+μ₂Δγ(n+1), where μ₁ andμ₂ are scaling factors defining a phase adjustment step size for eachsuccessive data symbol estimate, ψ is a phase slope indicative ofunbounded phase error due to frequency offset, and Δγ(n+1) is theminimum variance phase error estimate.
 19. The coherent receiver ofclaim 16, wherein the filter block comprises any one or more of: aWiener filter; and an averaging circuit configured to compute a runningaverage over the probabilistic phase error estimates.